Theorem brcogw 4780

Description: Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)

Assertion

Ref	Expression
brcogw	|- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> A ( C o. D ) B )

Proof of Theorem brcogw

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 995	. 2 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> A e. V )
2		simpl2 996	. 2 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> B e. W )
3		breq2 3993	. . . . . 6 |- ( x = X -> ( A D x <-> A D X ) )
4		breq1 3992	. . . . . 6 |- ( x = X -> ( x C B <-> X C B ) )
5	3, 4	anbi12d 470	. . . . 5 |- ( x = X -> ( ( A D x /\ x C B ) <-> ( A D X /\ X C B ) ) )
6	5	spcegv 2818	. . . 4 |- ( X e. Z -> ( ( A D X /\ X C B ) -> E. x ( A D x /\ x C B ) ) )
7	6	imp 123	. . 3 |- ( ( X e. Z /\ ( A D X /\ X C B ) ) -> E. x ( A D x /\ x C B ) )
8	7	3ad2antl3 1156	. 2 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> E. x ( A D x /\ x C B ) )
9		brcog 4778	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) ) )
10	9	biimpar 295	. 2 |- ( ( ( A e. V /\ B e. W ) /\ E. x ( A D x /\ x C B ) ) -> A ( C o. D ) B )
11	1, 2, 8, 10	syl21anc 1232	1 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> A ( C o. D ) B )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   E.wex 1485   e. wcel 2141   class class class wbr 3989   o. ccom 4615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-co 4620

This theorem is referenced by: (None)